Parametric Compression of Rank-1 Analog Feedback

ABSTRACT

Channel state information in a closed-loop, multiple-input, multiple-output wireless networks is fed back from each mobile station to a base station by first determining a transmit covariance matrix R, and applying a singular value decomposition (SVD) R=UΣV H , where U, V are left and right singular vector matrices, Σ is a diagonal matrix with singular values. The matrix V includes column vectors V. A beamforming vector v max =[1 exp(jΦ)exp(j2Φ) . . . exp(jΦ)]/√{square root over (N)}] is approximated by the column vector V having a maximum magnitude, where Φ is a real number. Then, only the angle Φ is fed back using a phase modulation mapping of the components exp(jΦ) onto the associated subcarrier.

RELATED APPLICATION

This Non-Provisional U.S. application claims priority to U.S.Provisional Application 61/173,126, entitled “Parametric Compression ofRank-1 Analog Feedback,” filed Apr. 27, 2009, incorporated herein byreference.

FIELD OF THE INVENTION

This invention relates generally to feeding back channel stateinformation (CSI), and more particularly to feeding back CSI inclosed-loop (CL) multiple-input, multiple-output (MIMO) wirelessnetworks.

BACKGROUND OF THE INVENTION

Feedback of channel stat information (CSI) can increase the performanceof closed-loop (CL) multiple-input, multiple-output (MIMO) wirelessnetworks. In MIMO networks, each cell includes a base station (BS) and aset of mobile stations (MS), where each MS estimates and feeds back theCSI of a downlink (DL) from the BS to the MS on an uplink (UL). Thefeedback can either be codebook based, or in analog form. The CSIfeedback is particularly important for the MS at edges of adjacent cellswhere inter-cell interference (ICI) can occur. As defined herein,feeding back means transmitting from the MS to the BS.

Feeding back rank adapted channel singular vectors, in a similar fashionto codebook based feedback, is superior to all other analog feedbackforms known for single-user (SU) MIMO networks.

Feedback in analog form mitigates colored interference regardless ofchannel conditions, signal-to-noise ratios (SNR) and interference color,when compared to codebook based feedback when both use the UL channelresources.

In addition to pure analog feedback options, one can also use analogfeedback for a differential mode. The difference is between the optimalsingular vector and the best codeword that is fed back unquantized.

The main appeal of analog feedback is for multi user MIMO (MU-MIMO)applications, or multi-BS MIMO applications, including femto and relaynetworks, where joint processing of multiple BS is performed to achievecoherent combining and interference nulling for MS at the edges of theadjacent cells.

This is mainly due to the fact that the channel feedback accuracy mustincrease linearly in dB with an increase in the SNR to remain withinfixed amount of dB from the MU-MIMO channel capacity.

Analog feedback is best suited for this task. The accuracy naturallyincreases with the SNR, and can provide simple and unified feedback fora typical macro-cell, as well as femto and pico cells, and relay stationto base station links, where the typical SNR is expected to be muchhigher.

On the other hand, codebook feedback in networks designed according tothe IEEE 802.16m standard limits the performance, due to channelfeedback quantization errors at the MS.

The current IEEE 802.16m “System Description Document” (SDD) assumesrank-1 feedback for MU-MIMO. Therefore, it is desired to optimize analogfeedback by informing the transmitter with a largest singular vector. Asdefined herein, the largest singular vector is the vector associatedwith the singular value with maximum magnitude.

General Analog Rank-1 Feedback

As shown in FIG. 1, for a conventional BS with N antennas, N complexvalued numbers are needed to represent the largest singular vector ofthe transmit covariance matrix. Therefore, at least N subcarriers areneeded to feedback a singular vector. That is, the transmit covariancematrix, which is represented by a matrix R 120, can be decomposed usinga singular value decomposition (SVD) as

R=UΣV^(H),

where U, V are the left and right singular vector matrices with N×Nentries, and Σ is an N×N diagonal matrix whose entries are the singularvalues.

A largest singular vector 110, i.e., the column of the matrix V with amaximum magnitude, is feedback to the BS. The components of the largestsingular vector {V₁, V₂, . . . , V_(N)} are assigned to N subcarriersassociated with N antennas. The complex numbers in the vector can bemapped to N subcarriers using, for example, amplitude modulation (AM).

Repetition 130 can be used to improve reliability in a low SNR range.Increasing the number of BS antennas improves performance on the UL. Ina BS with 4 or 8 antennas, no repetition is required for most SNRranges. Throughout this description, we use the notation V_(max), todenote the largest (maximum magnitude) singular vector.

It is possible to feedback only N−1 complex numbers by rotating allelements by a negative of an angle of the first element. This makes thefirst element real, so that the first element does not need to betransmitted.

If the angle of the vector V₁ is φ, then the feedback is

{exp(−jφ)*v₂, . . . , exp(−jφ)*v_(N)}, for j=1 to N.

At the BS, the first element can be determined because the sum power ofall elements is 1. However, this makes the feedback more sensitive topower normalizations.

Because feedback in cellular networks, and in particular networksaccording to the IEEE 802.16m standard, is done per band (that is over 1to 4 Physical Resource Blocks (PRB)). Two possibilities provide goodresults. Determine the largest singular vector of the average transmitcovariance matrix in that band. Computation of the transmit covariancematrix is simple and needed for the adaptive mode.

General computation of the largest singular vector can be facilitated inmost cases using a power method, or via the general SVD.

In most cases, the mobile stations have two receive antennas. Therefore,a simple closed form formula for the SVD of each subcarrier channel, forany number N of BS antennas, can be determined. It is also possible toaverage the singular vectors in a given band.

In particular for rank-1 feedback, the largest singular vector is simplya linear combination of the two channel rows, and the singular vectorsare phase aligned before being averaged across the band of interest. Asecond averaging iteration can improve performance.

For MS with four antennas, a maximum likelihood (ML) receiver can beused for spatial multiplexing. An implementation can use a spheredecoder. The sphere decoders performs a QR decomposition to construct asearch tree. After the QR decomposition is performed, obtaining the SVDis just another stage of the same procedure.

SUMMARY OF THE INVENTION

Channel state information in a closed-loop, multiple-input,multiple-output wireless networks is fed back from each mobile stationin a cell to a base station by first determining a transmit covariancematrix R, and applying a singular value decomposition (SVD) R=UΣV^(H),where U, V are left and right singular vector matrices, E is a diagonalmatrix with singular values. The matrix V includes column vectors V. Abeamforming vector v_(max)=[1 exp(jΦ)exp(j2Φ) . . . exp(jΦ)]/√{squareroot over (N)}] is approximated by the column vector V having a maximummagnitude, where an angle Φ is a real number. Then, only the angle Φ isfed back using a phase modulation mapping of the components exp(jΦ) ontothe associated subcarrier.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic of general prior art analog rank-1 feedback;

FIG. 2A is a schematic of a multi-input, multi-output network accordingto embodiments of the invention;

FIG. 2B is a schematic of analog feedback according to embodiments ofthe invention;

FIG. 3 is a schematic of feedback for a cross polarized array accordingto embodiments of the invention; and

FIG. 4 is a timing diagram a feedback method according to embodiments ofthe invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 2A shows a network according to embodiments of our invention. Acell 201 includes a base station (BS) 202, and a set of mobile stations(MS) 103. The BS communicates with each MS 203 via a downlink channel(DL) 204 using N antennas and N associated subcarriers, and the MS usean uplink channel (UL) 205 to communicate with the BS.

FIG. 2A also shows a portion of a potentially adjacent cell 207, whichcan cause inter-cell interfere (ICI) with the stations in the cell 101,particularly when a mobile station 108 is located near an edge of thecell.

Parametric Compression of Analog Rank-1 Feedback

In practical networks, closely spaced antennas are likely to be used toreduce costs of the BS. The antennas can be placed in a radar dome(radome). Neighborhood restrictions and zoning laws can also requiresmall antenna footprints.

Other advantages of closely spaced calibrated arrays stem from anincreased antenna correlation, which reduces a variability, acrossfrequency and time, of the spatial signature of the MS, and enablesimproved multi-user (MU) MIMO, and multi-BS MIMO performance with asmaller channel feedback overhead.

In general, we are concerned with feedback in analog form to supportrank-1 transmission. That is, the BS transmits a single data stream toeach MS using the N antennas available at the BS. The goal of the analogfeedback is to allow the BS to precode the data so that the MS receivesa signal with larger signal to interference and noise ratio (SINR) thanif the transmitter had no channel knowledge.

A transmit covariance matrix R is decomposed using a singular valuedecomposition (SVD) as

R=UΣV^(H),

where U, V are the left and right singular vector matrices with N×Nentries, the matrix V includes columns V, and H is a Hermitian operatorand Σ is an N×N diagonal matrix whose entries are the singular values.

In this section, we describe how knowledge of the geometry of thetransmit antenna array can be utilized to further reduce the amount offeedback that is necessary to give the transmitter an approximation ofthe largest singular vector. Because we are considering only rank-1transmission, the BS can beamform or “steer” transmission in thedirection of intended MS. Also, because particular array configurationsand geometries are assumed, we can make use of parametric expressionsfor the beamforming vectors as described below.

The following antenna configurations and feedback structures aredescribed and are applicable for the serving cell, as well as adjacentpotentially interfering cells (for multi-BS MIMO), i.e., λ/2 spaced 2, 4or 8 vertically polarized antenna arrays

-   -   (∥ or ∥∥ or ∥∥∥∥),        with the N antennas arranged in a uniform linear array (ULA),        and wherein a spacing between each antenna is at least half the        carrier wavelength λ.

In this case, the largest singular vector V, which is the vector with amaximum magnitude in the matrix V is approximated as a beamformingvector

v _(max)=exp(j(0:N−1)*Φ)/√{square root over (N)},

and only one real number Φ, which represents the angle of thebeamforming vector is fed back by using a phase modulation mapping ofexp(jΦ) onto the associated subcarrier. In conventional mathematicalnotation, exp(•)=e^((jΦ)). and colon, ‘:’, in general, (i: j is used tworepresent a sequence of integers [i, i+1, i+2, . . . , j], thus theequation uses a shorthand notation for the sequence of integers.

From the above equation, we see that the beamforming vector depends onthe single parameter, the angle Φ, and only single real value needs tobe fed back. That is, instead of feeding back the entire vector {V₁, V₂,. . . , V_(N)} on N subcarriers as in the prior art, we only feed back aparametric model of the singular vector as a steering vector. In thisway, all we feedback is the angle Φ.

FIG. 2B shows generalized feedback vector 210

V _(MAX)=[1exp(jΦ)exp(j2Φ) . . . exp(jΦ)]/√{square root over (N)}

for one of the N subcarriers 220 where j is an index for N, and anoptional repeated feedback 230.

Compressed Feedback for Cross Polarized Antenna Arrays

Alternatively, we can use cross polarized antennas. In this case, thespacing between antennas is still λ/2, and we have either two or fourcross polarized antennas

-   -   (XX or XXXX).

Another array geometry of four cross polarized antennas is two widelyspaced apart antennas.

For this case, the largest singular vector is approximated as twobeamforming vectors, one for each polarization, with gain and phaseoffset between the two vectors applied to the first beamforming vector.In this case, the beamforming vector can be expressed as V_(max)=[rexp(jθ)exp(j*(0:N/2−1)*(Φ₁)exp(j*(0:N/2−1)*Φ₂)]/√{square root over(N(1+r²)/2)}.

In the case of polarized antennas, we make a note of the dimensionalityof the singular vectors for each antenna geometry. Because each antennatransmits in two polarization, we actually have two logical elements foreach cross polarized antenna. Thus, for an array with two crosspolarized antennas, we have four antennas that can be used by thetransmitter. In the above equation, the parameter N is used to index theantennas. However, each of the polarization can only have a vector ofsize N/2 because each polarization sees N/2 antenna ports. Thus in theabove equation, the vector V_(max) has dimension N and each of the twobeamforming vectors for each polarization has dimension N/2.

The feedback includes one complex value, r exp(jθ), which is the gainand phase offset between the two vectors, and which is mapped to onesubcarrier (r=1 on average), and two real values, Φ₁ and Φ₂, which aremapped to two subcarriers.

FIG. 3 shows the feedback of the vector V_(max) 310 for the crosspolarized array. The parameters r exp(jθ) 311, Φ₁ 312, and Φ₂ 313 aremapped to three subcarriers 301, 302, 303. An optional repeated feedback330 is also possible.

Alternatively, we can use two cross polarized antennas (XX), spacedapart by λ/2. In this case, further compression of the above feedbackstructure is possible with only a small decrease in performance. Giventhat there are only two closely spaced antennas, inaccuracies in theparameters Φ₂ or Φ₁ reduce performance degradation. Hence, we canamplitude modulate the subcarrier carrying exp(jΦ₁) by a real parameterc. This parameter c is proportional to a deviation of Φ₂ from Φ₁ andequals 1 when Φ₂=Φ₁. On average, this deviation is zero as the long termangle of departure (AoD) for either polarization is the same. Thefeedbacks r exp(jθ) 311 and c exp(jΦ₁) 312 are mapped onto twosubcarriers 310-302. and the largest singular vector is approximated as

V _(max) ≈[rexp(jθ)exp(j*(0:1)*Φ₁)exp(j*(0:1)*(Φ₁+(c−1))]/√{square rootover (2+2r ²)}.

In yet another configuration, we consider two widely spaced clusters ofantenna Configuration 3 (XX XX), described below.

This antenna configuration is practical because this configurationreplaces the typical two-antenna diversity configuration in many cellsites. The feedback is now done on five subcarriers.

Four real phases corresponding to the four λ/2 spaced groups ofantennas, which are mapped onto two subcarriers as in Configuration 3:c₁ exp(jΦ₁) and c₂ exp(jΦ₂).

Three complex numbers α₁ α₂ α₃ corresponding to the gain and phaseoffset of three groups relative to the first beamforming vector.

The largest singular vector is approximated as

$V_{\max} \cong {\begin{bmatrix}{1{\exp \left( {j\Phi}_{1} \right)}} \\{\alpha_{1}\alpha_{1}{\exp \left( {j\left( {\Phi_{1} + c_{1} - 1} \right)} \right)}} \\{\alpha_{2}\alpha_{2}{\exp \left( {j\left( \Phi_{2} \right)} \right)}} \\{\alpha_{3}\alpha_{3}{\exp \left( {j\left( {\Phi_{2} + c_{2} - 1} \right)} \right)}}\end{bmatrix}/\sqrt{2 + {2{\sum\limits_{i = 1}^{3}\; {\alpha_{i}}^{2}}}}}$

Four antenna configurations classes are defined with mappings to one,three or five subcarriers for eight antennas, and one or two subcarriers for four antennas. In all cases, the feedback overhead fitsinto half of a Feedback Mini-Tile (FMT) as defined in the current systemdescription document (SDD) of the IEEE 802.16m standard, and repetitioncoding with mapping onto different FMTs belonging to the same SecondaryFast Feedback Control Channel (SFBCH) can be applied for improvedperformance at low SNR.

The MS can detect the amount of antenna correlation using the long termtransmit covariance matrix used for the adaptive MIMO mode, and the MAScan signal whether a compressed mode feedback is used.

The BS broadcasts the antenna configuration to facilitate thisoperation, wherein correlated antennas are allocated consecutivenumbers. In this case, if the BS uses four λ/2 spaced cross polarizedantennas, then antennas 1-4 are used for a first polarization, andantennas 5-8 for a second. Without loss of generality, this order ofantennas is used to index the antenna when cross polarizedconfigurations are considered.

The overall method is shown in FIG. 4 with time running down. The BSantenna configuration is broadcast 410 to the MS, and the MS. Thebroadcast message can include information on the antenna configurationsof neighboring BS. Based on this message, the BS request 420 the MS touse uses an appropriate parametric feedback compression technique 425based on the BS antenna configuration, and the MS can feedback 430 abest feedback compression for a particular antenna configuration.

While the implementation of the method is network specific, following isa description of the various feedback techniques can be used for variousantenna configuration.

Antenna Configuration 1—One Beamforming Vector

This case corresponds to the ULA with N closely spaced transmit antennasat the BS. We describe three methods to estimate an angle for thebeamforming vector

Assuming the optimal singular vector of column V is determined asdescribed above, we can estimate

${{\exp ({j\Phi})} = \frac{x}{x}},$

where x=V(2:N)^(H)*V(1:N−1).

Using the transmit covariance matrix R, the objective is to determinethe beamforming vector V_(max) that maximizes V^(H)RV. If we denote theN components of the vector V_(max) by

e^(jΦ) ^(i) , i=1, 2, . . . N,

then, we obtain

$\begin{matrix}{{V^{H}{RV}} = {\sum\limits_{m}^{\;}\; {\sum\limits_{n}^{\;}\; {R_{mn}^{{j\Phi}{({m - n})}}}}}} \\{{= {2{Re}\left\{ {\sum\limits_{k = 0}^{N - 1}\; {S_{k}^{{j\Phi}\; k}}} \right\}}},}\end{matrix}$

where the function Re{x} is the real part of x, and

$S_{k} = {\sum\limits_{n}^{\;}\; R_{n + {kn}}}$ for  k > 0 and${S_{0} = {0.5{\sum\limits_{n}^{\;}\; R_{nn}}}},$

where R_(mn) is an entry at the m^(th) row and n^(th) column of thematrix R.

The value Φ that maximizes the above expression can be found to anydegree of accuracy by taking a fast Fourier transform (FFT) of the Nvalues S_(k).

Alternatively, we can assume that the MS and BS share a set ofpredefined beamforming vectors {V_(s)}. Using the transmit covariancematrix R, the MS can perform an exhaustive search of the beamformingvector that maximizes V_(s) ^(H)RV_(s).

Then the feedback includes the index of the maximizing V_(s). A searchspace of 64 options provides very good performance for eight antennas.Thus, with this method, six bits (6=log₂(64)) can be fed back to the BS.Using the transmit covariance matrix R, an exhaustive search of thebeamforming vector that maximizes V^(H)RV can be performed.

Antenna Configuration 2—Two Beamforming Vectors

This case corresponds to arrays with two or four cross polarizedtransmit antennas, where each antenna is spaced at least half awavelength apart at the BS. We provide three methods to estimate theangles for the two required beamforming vectors.

Assuming the optimal singular vector column V is determined as describedabove, we estimate

${\exp \left( {j\Phi}_{1} \right)} = {\frac{x}{x}.}$

where

x=V(2:N/2)^(H) *V(1:N/2−1),

and

${{\exp \left( {j\Phi}_{2} \right)} = \frac{x}{x}},$

where

x=V(N/2+2:N)^(H) *V(N/2+1:N−1).

These estimates are used to estimate the gain and phase offset as

${{r\; {\exp ({j\theta})}} = \frac{{V\left( {1\text{:}\mspace{14mu} {N/2}} \right)}^{{{j\Phi}_{1}{\lbrack{0\text{:}3}\rbrack}}^{\prime}}}{{V\left( \; {{N/2} + {1\text{:}\mspace{14mu} N}} \right)}^{{{j\Phi}_{2}{\lbrack{0\text{:}3}\rbrack}}^{\prime}}}},$

and

Using the transmit covariance matrix R, we first estimate the gain as

${r = {{sqrt}\left( \frac{\sum\limits_{i = 1}^{N/2}\; R_{ii}}{\sum\limits_{i = 1}^{N}\; R_{ii}} \right)}},$

and then assuming the two beamforming vectors are known, the phaseoffset is estimated using the top right quadrant of the matrix R, whichrepresents the cross talk between the polarizations. In this case, thematrix R has the following form

${R = \begin{bmatrix}Q_{1} & Q_{3} \\Q_{3}^{H} & Q_{2}\end{bmatrix}},$

where Q_(i), i=1, 2, 3 are the quadrants of the matrix R.

An expression

${{\exp ({j\theta})} = \frac{x}{x}},$

where θ is the phase offset between the beamforming vectors, can bedetermined as

x=e ^(−jΦ) ¹ ^([0:N/2−1]) Q ₃ e ^(jΦ) ² ^([0:N/2−1]′).

The two beamforming vectors can be determined separately by maximizingeach vector with its respective quadrant

e ^(−jΦ) ^(i) ^([0:3]) Q _(i) e ^(jΦ) ^(i) ^([0:3]′)

as was done in the case of one beamforming vector, or jointly byexhaustive maximization of V^(H)RV where

V=[rexp(jθ)exp(j*(0:N/2−1)*Φ₁)exp(j*(0:N/2−1)*Φ₂)]/√{square root over(N(1+r ²)/2)}.

Antenna Configuration 3—Compressed Two Beamforming Vectors for XX

This procedure is similarly to antenna Configuration 2 with theestimation

c=√{square root over (1+sin(Φ₂−Φ₁))}.

The square root operation is intended to compress the amplitude closerto 1.

Antenna Configuration 4—Four Beamforming Vectors:

Assuming the optimal singular vector column V is determined as describedabove, the four beamforming vectors per group and the three complexratios can be determined according to the steps of antenna Configuration3.

Alternative Mapping Approach

While the mapping of the parameters is done in an analog manner by usingsimple AM and phase modulation (PM), other mappings are possible. Forexample, the parameters can be digitized, and the first n (n=2 or more)most significant bits of the parameters can be transmitted digitally ona control channel.

Adaptive Mode

Similarly to the codebook approach, wherein a long term transmitcorrelation matrix is used in the adaptive mode to transform thecodebook and improve performance, we can improve the performance of thecompressed analog feedback mode.

The long term transmit correlation matrix used in the adaptive MIMO modeis denoted as R_(LT), which is averaged over multiple sub-carriers. Bydenoting R as the for the narrow band transmit correlation matrix, whichis estimated over one or a small number of PRB, the general objectivebecomes of determining the vector V that maximizes

$\frac{V^{H}R_{LT}^{H}{RR}_{LT}V}{{{R_{LT}V}}^{2}},$

where V is of one of the parameterized structures as described above.

Although the invention has been described by way of examples ofpreferred embodiments, it is to be understood that various otheradaptations and modifications can be made within the spirit and scope ofthe invention. Therefore, it is the object of the appended claims tocover all such variations and modifications as come within the truespirit and scope of the invention.

1. A method for feeding back channel state information (CSI) in aclosed-loop (CL) multiple-input, multiple-output (MIMO) wirelessnetworks, wherein the network includes a plurality of cells, and eachcell includes a base station (BS) and a set of mobile stations (MS),wherein each BS includes a set of N antennas spaced apart by at leasthalf a carrier wavelength λ, and wherein the N antennas are associatedwith N subcarriers, comprising at each MS: determining a matrix R,wherein the matrix R is a transmit covariance matrix; applying asingular value decomposition (SVD) to the matrix R according toR=UΣV^(H), where U, V are left and right singular vector matrices, Σ isa diagonal matrix with singular values, the matrix V includes columnvectors V, and ^(H) is a Hermitian operator; approximating a beamformingvector v_(max) by the column vector V having a maximum magnitude, andwhereinv _(max)=[1exp(jΦ)exp(j2Φ) . . . exp(jΦ)]/√{square root over (N)}],where an angel Φ is a real number, and j indexes N; and feeding back, inanalog form, from the MS to the BS, only the angle Φ using a phasemodulation mapping of components exp(jΦ) the vector v_(max) onto theassociated subcarrier.
 2. The method of claim 1, further comprising:repeating the feeding back
 3. The method of claim 1, wherein the BSincludes multiple cross polarized antennas and whereinv _(max) =[rexp(jθ)exp(j*(0:N/2−1)*Φ₁)exp(j*(0:N/2−1)*Φ₂)]/√{square rootover (N(1+r ²)/2)}, wherein the complex value r exp(jθ) is a gain and aphase offset between two beamforming vectors mapped to one subcarrier,and wherein Φ₁ and Φ₂ are mapped to two other subcarriers.
 4. The methodof claim 3, wherein the BS includes two cross polarized antennas spacedby λ/2, and further comprising: amplitude modulating the subcarriercarrying exp(jΦ₁) by a real parameter c wherein the parameter c isproportional to a deviation of Φ₂ from Φ₁ and equals 1 when Φ₂=Φ₁. 5.The method of claim 4, whereinv _(max) ≈[rexp(jθ)exp(j*(0:1)*(Φ₁)exp(j*(0:1)*(Φ₁+(c−1))]/√{square rootover (2+2r ²)}, wherein the notation (0:1) represents a sequence ofintegers [i, i+1, i+2, . . . , j].
 6. The method of claim 1, wherein theBS includes widely spaced clusters of antennas, and the feeding backmaps c₁ exp(jΦ₁) and c₂ exp(jΦ₂) to two subcarriers, and three complexnumbers α₁ α₂ α₃ correspond to the gain and the phase offset, and$\begin{bmatrix}{1{\exp \left( {j\; \Phi_{1}} \right)}\alpha_{1}\alpha_{1}{\exp \left( {j\left( {\Phi_{1} + c_{1} - 1} \right)} \right)}} \\{\alpha_{2}\alpha_{2}{\exp \left( {j\left( \Phi_{2} \right)} \right)}} \\{\alpha_{3}\alpha_{3}{\exp \left( {j\left( {\Phi_{2} + c_{2} - 1} \right)} \right)}}\end{bmatrix}/\sqrt{2 + {2{\sum\limits_{i = 1}^{3}\; {\alpha_{i}}^{2}}}}$7. The method of claim 1, further comprising: signaling to the BSwhether the MS is using a compressed mode feedback.
 8. The method ofclaim 1, further comprising: broadcasting, by the BS, an antennaconfiguration used by the BS.
 9. The method of claim 8, furthercomprising: broadcasting by the BS antenna configurations of neighboringBS.
 10. The method of claim 9, wherein the feeding back is for aparticular antenna configuration.
 11. The method of claim 1, wherein${{\exp \left( {j\; \Phi} \right)} = \frac{x}{x}},$ wherex=V(2:N)^(H) *V(1:N−1), and the notation (i:j) represents a sequence ofintegers [i, i+1, i+2, . . . , j].
 12. The method of claim 1, whereinthe column vector in the matrix v having the maximum magnitude maximizesV^(H)RV, and the components of the vector V_(max) for the N antennas aree^(jΦ) ^(i) , i=1, 2, . . . , N, and${{V^{H}{RV}} = {{\sum\limits_{m}\; {\sum\limits_{n}{R_{mn}^{j\; {\Phi {({m - n})}}}}}} = {2\; {Re}\left\{ {\sum\limits_{k = 0}^{N - 1}\; {S_{k}^{j\; \Phi \; k}}} \right\}}}},$where the notation (i:j) represents a sequence of integers [i, i+1, i+2,. . . , j], the function Re{x} is the real part of x, and$S_{k} = {\sum\limits_{n}R_{n + {kn}}}$ for  k > 0  and  ${S_{0} = {0.5{\sum\limits_{n}R_{nn}}}},$ where R_(mn) is an entry atthe m^(th) row and n^(th) column of the matrix R.
 13. The method ofclaim 12, wherein Φ is maximized by taking a fast Fourier transform(FFT) of the N values of S_(k).
 14. The method of claim 2, wherein${\exp \left( {j\; \Phi_{1}} \right)} = {\frac{x}{x}.}$ wherex=V(2:N/2)^(H) *V(1:N/2−1), where the notation (i:j) represents asequence of integers [i, i+1, i+2, . . . , j], and${{\exp \left( {j\; \Phi_{2}} \right)} = \frac{x}{x}},$ wherex=V(N/2+2:N)^(H) *V(N/2+1:N−1), and the gain is${{r\; {\exp \left( {j\; \theta} \right)}} = \frac{{V\left( {1\text{:}\mspace{14mu} {N/2}} \right)}^{j\; {\Phi_{1}{\lbrack{0\text{:}3}\rbrack}}^{\prime}}}{{V\left( {{N/2} + {1\text{:}\mspace{14mu} N}} \right)}^{{{j\Phi}_{2}{\lbrack{0\text{:}3}\rbrack}}^{\prime}\;}}},$and${r = {{sqrt}\left( \frac{\sum\limits_{i = 1}^{N/2}\; R_{ii}}{\sum\limits_{i = 1}^{N}\; R_{ii}} \right)}},$and the phase offset is estimated using a top right quadrant of thematrix R ${R = \begin{bmatrix}Q_{1} & Q_{3} \\Q_{3}^{H} & Q_{2}\end{bmatrix}},$ which represents the cross talk between thepolarizations, where Q_(i), i=1, 2, 3 are quadrants of the matrix R, and${{\exp \left( {j\; \theta} \right)} = \frac{x}{x}},$ where θ isthe phase offset between the beamforming vectors isx=e ^(−jΦ) ¹ ^([0:N/2−1]) Q ₃ e ^(jΦ) ² ^([0:N/2−1]′).
 15. The method ofclaim 4, whereinc=√{square root over (1+sin(Φ₂−Φ₁))}.
 16. The method of claim 1, whereinthe matrix R is a long term transmit correlation matrix R_(LT), which isaveraged over multiple sub-carriers, and wherein the column vector Vhaving the maximum magnitude maximizes$\frac{V^{H}R_{LT}^{H}{RR}_{LT}V}{{{R_{LT}V}}^{2}}.$